Errors When Shock Waves Interact Due to Numerical Shock Width

A simple test problem proposed by Noh, a cold gas with uniform initial particle velocity directed towards a rigid wall, demonstrates a generic problem with numerical shock capturing algorithms at boundaries that Noh called "excess wall heating" The same type of numerical error is shown to occur when shock waves interact. The underlying cause is due to the numerical shock profile. The error can be understood from an analysis of the asymptotic solution of the partial differential equations when an artificial viscosity is added. The position of the front for a numerical shock wave can be defined by matching the total mass in the profile to that of a discontinuous shock. There is then a difference in the total energy of the numerical wave relative to a discontinuous shock. Moreover, the relative energy depends on the strength of the shock. The error when shock waves interact results from the difference in the relative energies between the incoming and outgoing shock waves. A conservative differencing scheme correctly describes the Hugoniot jump conditions for a steady propagating shock. The error implied by the asymptotic energy shift occurs in the entropy generated by the numerical dissipation in the transient when the waves interact. The entropy error remains localized and does not dissipate. A scaling argument shows that as the viscosity coefficient approaches zero, the error shrinks in spatial extent but the peak pointwise error is constant in magnitude. Consequently, the convergence of the inviscid limit to the hyperbolic solution is nonuniform in regions where shocks have interacted.